Verify $H_2(x) = 4x^2-2$ satisfies Hermite differential equation.

Question

I am slightly confused by the meaning of the question:

Q. The differential equation obeyed by Hermite polynomial is

$$y''-2xy'+2ny=0$$

a) Use the Rodrigues' formula for Hermite polynomial

$$H_n(x)=(-1)^n e^{x^2}\dfrac{d^n}{dx^n}e^{-x^2}$$

to find $H_2(x)$ and verify that it satisfies the above differential equation.

So by doing this I got $H_2(x) = 4x^2-2$. But what dose it mean by verify? How should I approach this?

Answer 1

\begin{eqnarray*} H_2(x)=\color{orange}{4x^2-2} \\ H_2'=\color{red}{8x} \\ H_2''=\color{purple}{8} . \end{eqnarray*} Substitute into $y''-2xy'+2ny$ and verify that it gives zero, \begin{eqnarray*} \color{purple}{8} -2x(\color{red}{8x} )+2\times 2( \color{orange}{4x^2-2}) =0. \end{eqnarray*}